In a groundbreaking development, a mathematician from UNSW Sydney, Honorary Professor Norman Wildberger, has unveiled a novel method to address one of algebra's oldest challenges—solving higher polynomial equations. Polynomials are fundamental equations that involve a variable raised to various powers, such as the degree two polynomial: 1 + 4x – 3x² = 0. These equations are not only vital in mathematics but also play a crucial role in science, helping to describe phenomena like planetary movement and aiding in computer programming.
Historically, finding a general method for solving higher-order polynomial equations, particularly those where x is raised to the fifth power or higher, has proven to be an elusive goal. Solutions to degree-two polynomials have existed since as early as 1800 BC, thanks to the Babylonians' method of completing the square, which eventually evolved into the familiar quadratic formula taught in high schools today. The methods for solving three- and four-degree polynomials emerged in the 16th century, and by 1832, French mathematician Évariste Galois demonstrated that the mathematical symmetry used for lower-order polynomials could not be applied to polynomials of degree five or higher, concluding that no general formula could exist for them.
Since Galois's findings, various approximate solutions for higher-degree polynomials have been developed and widely utilized in practical applications. However, Professor Wildberger argues that these approximations do not belong to the realm of pure algebra. He identifies the core issue as the reliance on classical formulas that use third or fourth roots, known as radicals. These radicals often represent irrational numbers, which are decimals that extend infinitely without repeating and cannot be represented as simple fractions. For example, the cubed root of seven, 3√7 = 1.9129118..., is a number that continues indefinitely.
Professor Wildberger asserts that the reliance on such irrational numbers leads to logical inconsistencies within mathematics. His skepticism towards irrational numbers stems from their dependence on an imprecise concept of infinity, which complicates mathematical reasoning. This rejection of radicals has significantly influenced his notable contributions to mathematics, particularly in the fields of rational trigonometry and universal hyperbolic geometry. Both of these frameworks emphasize mathematical operations such as squaring, adding, or multiplying instead of relying on irrational numbers.
In his innovative approach to solving polynomials, Professor Wildberger's method avoids radicals and irrational numbers entirely. Instead, it employs special extensions of polynomials known as power series, which can contain an infinite number of terms with varying powers of x. By truncating these power series, he and his collaborator, computer scientist Dr. Dean Rubine, were able to extract approximate numerical answers, effectively validating the method. One of the equations tested was a historical cubic equation used by Wallis in the 17th century to illustrate Newton's method, and the results were promising.
Ultimately, Professor Wildberger states that the proof of this new method is grounded in mathematical logic. The technique utilizes unique sequences of numbers that represent complex geometric relationships, belonging to the branch of mathematics known as combinatorics. One of the most famous sequences in combinatorics is the Catalan numbers, which quantify the various ways to dissect a polygon into triangles. These numbers have significant practical applications, including in computer algorithms, data structure designs, and game theory, and are even applicable in biological contexts, such as counting RNA folding patterns.
Professor Wildberger's innovative approach suggests that to solve higher-degree equations, one should seek higher analogs of the Catalan numbers. His research extends these numbers from a one-dimensional to a multi-dimensional array, based on the numerous ways a polygon can be divided using non-intersecting lines. This discovery has led to a substantial revision of fundamental algebra principles, providing solutions even for quintics, or degree five polynomials. Beyond theoretical implications, this method holds practical promise for developing computer programs that can solve equations using algebraic series rather than relying on radicals.
Professor Wildberger and Dr. Rubine have also introduced an innovative array of numbers called the Geode, which extends classical Catalan numbers and appears to underlie them. The exploration of this new Geode array is expected to spark numerous inquiries, keeping combinatorialists engaged for years to come. This research represents just the beginning of an exciting new chapter in mathematics, with vast potential for future discoveries and applications.
